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\title{《基础复分析》第5章复积分 - 课文}
\author{CGZ ET AL}

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\begin{document}
\maketitle 

\section*{5.1. Cauchy定理}

\begin{enumerate}

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\item % 5.1
设 $p,q$ 是平面区域 $\Omega$ 上的连续函数。
线积分 $\int_\gamma pdx+qdy$ 只依赖于曲线端点的充要条件是 $pdx+qdy$ 是全微分，
即存在函数 $G$ 使得 $dG = pdx+qdy$. 

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\item % 5.2
设 $f(z)$ 在包含闭矩形区域 $R$ 的一个区域上解析，则 $$\int_{\partial R}f(z)=0.$$

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\item % 5.3
设 $f(z)$ 在包含闭矩形区域 $R$ 的一个区域上除去 $R$ 内的有限个点 $\zeta_1,\cdots,\zeta_n$ 之外解析，
且对所有 $\zeta_j$ 都满足 $\lim\limits_{z\to\zeta_j} (z-\zeta_j)f(z)=0$, 则 $$\int_{\partial R}f(z)=0.$$

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\item % 5.4
设函数 $f(z)$ 在开圆盘 $\Delta$ 内解析，则对 $\Delta$ 内的任意闭曲线 $\gamma$, 有 
$$\int_\gamma f(z)=0.$$

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\item % 5.5
设函数 $f(z)$ 在开圆盘 $\Delta$ 内除去 $R$ 内的有限个点 $\zeta_1,\cdots,\zeta_n$ 之外解析，
且对所有 $\zeta_j$ 都满足 $\lim\limits_{z\to\zeta_j} (z-\zeta_j)f(z)=0$, 
则对 $\Delta$ 内的任意不通过 $\zeta_j$ 的闭曲线 $\gamma$, 有 $$\int_\gamma f(z)=0.$$

\end{enumerate}

\section*{5.2. Cauchy积分公式}

\begin{enumerate}[start=6]
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\item % 引理5.1
设 $a$ 为平面上一点，$\gamma$ 是平面上一条不通过点 $a$ 的闭曲线，则积分 $$\frac{1}{2\pi i}\int_\gamma \frac{dz}{z-a}$$ 为整数。称为闭曲线 $\gamma$ 关于点 $a$ 的环绕数，记为 $n(\gamma,a)$. 

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\item % 5.6
设函数 $f(z)$ 在开圆盘 $\Delta$ 内解析，设 $\gamma$ 是 $\Delta$ 内的一条闭曲线, 则对不在 $\gamma$ 上的任意点 $z$, 有  
$$n(\gamma,z)f(z) = \frac{1}{2\pi i}\int_\gamma \frac{f(\zeta)d\zeta}{\zeta-z}.$$

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\item % 5.7
设函数 $f(z)$ 在开圆盘 $\Delta$ 内除去有限个点 $\zeta_1,\cdots,\zeta_n$ 之外解析，
且对所有 $\zeta_j$ 都满足 $\lim\limits_{z\to\zeta_j} (z-\zeta_j)f(z)=0$. 
设 $\gamma$ 是 $\Delta$ 内不通过所有 $\zeta_j$ 的一条闭曲线，
则对不在 $\gamma$ 上的任意点 $z$, 只要 $z\neq \zeta_j$, 都有  
$$n(\gamma,z)f(z) = \frac{1}{2\pi i}\int_\gamma \frac{f(\zeta)d\zeta}{\zeta-z}.$$

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\item % 5.8
设 $f(z)$ 是区域 $\Omega$ 内的解析函数，则 $f(z)$ 的各阶导数都存在。特别地，设 $C$ 是 $\Omega$ 内的一个圆周，使得 $C$ 的内部 $\Delta$ 也包含在 $\Omega$ 内。则对 $z\in \Delta$, 有
$$f^{(n)}(z) = \frac{n!}{2\pi i}\int_C \frac{f(\zeta)d\zeta}{(\zeta-z)^{n+1}}.$$

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\item % Morera
设 $f(z)$ 是区域 $\Omega$ 内的连续函数，如果对 $\Omega$ 内的所有闭曲线 $\gamma$, 都有 $\int_\gamma f(z)dz=0$, 则 $f(z)$ 在 $\Omega$ 内解析。

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\item % Liouville
全平面上的有界解析函数必是常数。

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\item % 代数基本定理
非常数多项式必有零点。

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\item % 5.9
设 $f(z)$ 在区域 $\Omega$ 内除去一点 $a\in\Omega$ 之外解析。则存在 $\Omega$ 内的解析函数 $g(z)$ 使得 $g(z)\mid_{\Omega\setminus \{a\}} = f(z)$ 的充分必要条件是 $a$ 是 $f(z)$ 的可去奇点。

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\item % 5.10 Taylor
设 $f(z)$ 在包含 $a$ 点的区域 $\Omega$ 内解析。则有 
$$ f(z) = f(a) + f'(a)(z-a) + \frac{f''(a)}{2!}(z-a)^2 + \cdots + \frac{f^{(n-1)}(a)}{(n-1)!}(z-a)^{n-1} 
+ f_n(z)(z-a)^n,$$
其中 $f_n(z)$ 在 $\Omega$ 内解析。

\end{enumerate}

\section*{5.3. 解析函数的局部性质}

\begin{enumerate}[start=15]
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\item % 5.11
非常数解析函数的零点是孤立的。即：设 $f(z)$ 和 $g(z)$ 是区域 $\Omega$ 上的两个解析函数。
如果存在子集 $E\subset\Omega$ 使得在 $E$ 上有 $f(z)=g(z)$, 且 $E$ 在 $\Omega$ 内有聚点，则在 $\Omega$ 内有 $f(z)=g(z)$. 

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\item % 5.12
解析函数在本性奇点的一个邻域内可以逼近任意复数。

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\item % 5.13
设 $f(z)$ 是开圆盘 $\Delta$ 上的非常数解析函数，$\{a_i\}$ 是 $f(z)$ 在 $\Delta$ 内的所有零点，其阶为 $k_i$. 
设 $\gamma$ 为 $\Delta$ 内不通过零点的一条闭曲线。则 $$\sum_i k_i n(\gamma,a_i) = \frac{1}{2\pi i} \int_\gamma \frac{f'(z)}{f(z)}dz, $$ 其中的和式只有有限项不为零。

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\item % 5.14 局部对应
设 $f(z)$ 是区域 $\Omega$ 上的解析函数，$a\in\Omega$ 是 $f(z)-f(a)$ 的 $k$ 阶零点。
则当 $\varepsilon>0$ 足够小时，存在 $\delta>0$, 只要 $0<|B-f(a)|<\delta$, 方程 $f(z)=B$ 在圆盘 $|z-a|<\varepsilon$ 内恰好有 $k$ 个根，且都是单根。

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\item % 5.5. 开映射
非常数解析函数将开集映为开集。

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\item % 5.6 共形映射
如果解析函数 $f(z)$ 在 $a$ 点的导数不为零，则在 $a$ 点的一个邻域内 $f(z)$ 是共形映射。

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\item % 5.7
设 $f(z)$ 是区域 $\Omega$ 上的解析函数，$a\in\Omega$ 是 $f(z)-f(a)$ 的 $k(k\ge 2)$ 阶零点。
则存在 $a$ 点的一个邻域上的共形映射 $\zeta(z)$, 满足 $\zeta(a)=0$, 使得 $$f(z)-f(a)=\zeta(z)^k. $$

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\item % 5.15 最大模原理
非常数解析函数的模在区域内部没有最大值。

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\item % 5.15
设 $f(z)$ 是区域 $\Omega$ 上非常数的解析函数。则对 $a\in\Omega$, $$|f(a)|< \overline{\lim\limits_{z\to\partial\Omega}} |f(z)| := \underset{b\in\partial\Omega}{\sup}\overline{\lim\limits_{z\to b}} |f(z)|. $$

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\item % 5.16 施瓦茨引理
设 $f(z)$ 是单位圆盘 $D$ 上的解析函数，满足 $|f(z)|<1$ 且 $f(0)=0$. 
则 $|f(z)|\le |z|$, 且 $|f'(0)|\le 1$. 
等号在一点成立，当且仅当 $f(z)$ 是一个旋转，即 $f(z)=e^{i\theta}z, \theta\in\mathbb{R}$. 


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\item % 例5.5
证明圆盘之间的共形映射一定是分式线性变换。


\end{enumerate}

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\end{document}

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